An object is at rest if its position does not change with time with respect to its surroundings, and in motion if its position changes. Rest and motion are relative: a passenger sitting in a moving bus is at rest with respect to the bus but in motion with respect to the road. To describe the position of an object we need a reference point (origin). Motion can be along a straight line (rectilinear), circular, or periodic (repeating, like a pendulum).

Distance is the total length of the path travelled by an object; it is a scalar (only magnitude), always positive and never zero for a moving body. Displacement is the shortest straight-line distance from the initial to the final position, with direction; it is a vector and can be positive, negative or zero. SI unit of both: metre (m). Displacement can be less than or equal to the distance, never more. Example: if a person walks 3 m north and then 4 m east, distance = 7 m and displacement = 5 m. After a full round of a circular track, displacement is zero while distance is the circumference.

Speed is the distance travelled per unit time: speed = distance ÷ time. It is a scalar. SI unit: m/s (or m s⁻¹); also km/h. Conversion: 1 km/h = 5/18 m/s; so 72 km/h = 20 m/s. Average speed = total distance travelled ÷ total time taken. A body can have different speeds at different instants. Speedometer in a vehicle measures the speed; the odometer measures the distance travelled.

Velocity is the displacement per unit time, or the speed in a given direction: velocity = displacement ÷ time. It is a vector. SI unit: m/s. Average velocity = total displacement ÷ total time. For uniform motion, average velocity = (initial velocity + final velocity) ÷ 2 only when acceleration is uniform. Speed and velocity are equal in magnitude only when the body moves in a straight line without changing direction. Velocity changes if either speed or direction changes.

Acceleration is the rate of change of velocity: a = (v − u) ÷ t, where u is the initial velocity, v the final velocity and t the time. SI unit: m/s² (m s⁻²). It is a vector. If velocity increases with time, acceleration is positive; if velocity decreases, acceleration is negative and is called retardation or deceleration. Uniform acceleration: velocity changes by equal amounts in equal intervals of time (e.g. a freely falling body). Non-uniform acceleration: velocity changes by unequal amounts in equal intervals.

Uniform motion: the body covers equal distances in equal intervals of time, however small the intervals (constant speed in a straight line). Non-uniform motion: the body covers unequal distances in equal intervals of time. Example of uniform motion: a car moving at constant 60 km/h on a straight road. Example of non-uniform motion: a car in city traffic, a freely falling stone. In uniform circular motion the speed is constant but velocity changes continuously, so it is an accelerated motion.

A distance–time graph has time on the x-axis and distance on the y-axis. Straight line parallel to the time axis: the body is at rest. Straight line sloping upward: uniform speed; slope = speed. Curved line: non-uniform speed (a steeper curve means higher speed). A steeper straight line represents greater speed. The distance–time graph cannot slope downward, as distance never decreases. Speed from the graph = (change in distance) ÷ (change in time).

A velocity–time graph has time on the x-axis and velocity on the y-axis. Straight line parallel to the time axis: uniform velocity (zero acceleration). Straight line sloping upward: uniform acceleration; sloping downward: uniform retardation. Curved line: non-uniform acceleration. Slope of the v–t graph = acceleration. Area under the v–t graph = displacement. For a uniformly accelerated body starting from rest, the graph is a straight line through the origin; the area is a triangle, ½ × t × v.

For a body moving in a straight line with uniform acceleration a, initial velocity u, final velocity v, time t and displacement s: First equation: v = u + at. Second equation: s = ut + ½at². Third equation: v² = u² + 2as. Sign convention: take the direction of initial velocity as positive; retardation is negative. For a body starting from rest u = 0; for a body that comes to rest v = 0. These equations hold only for uniform acceleration along a straight line.

Take a v–t graph with a straight line from point A (initial velocity u at t = 0) to point B (final velocity v at time t). First equation: slope AB = a = (v − u) ÷ t, so v = u + at. Second equation: s = area under the graph = area of rectangle (u × t) + area of triangle (½ × t × (v − u)) = ut + ½t(at) = ut + ½at². Third equation: s = average velocity × time = ((u + v) ÷ 2) × t, with t = (v − u) ÷ a, so s = (v² − u²) ÷ 2a, i.e. v² = u² + 2as.

A body falling only under the Earth's gravity is in free fall. The acceleration due to gravity g is about 9.8 m/s² (taken as 10 m/s² in many numericals) directed downward, and does not depend on the mass of the body. For a body dropped from rest: v = gt, h = ½gt², v² = 2gh. For a body thrown upward, g is taken as negative: it slows down, becomes zero at the highest point, then falls back. Maximum height H = u² ÷ 2g and time of ascent = u ÷ g; the time of descent equals the time of ascent.

When a body moves along a circular path with constant speed, it is in uniform circular motion. Its speed is constant, but the direction of velocity changes continuously, so velocity changes and the motion is accelerated. Velocity at any point is along the tangent to the circle. Speed v = 2πr ÷ T, where r is the radius and T the time for one revolution. Examples: the Moon around the Earth, tip of a clock's hands, an athlete running on a circular track, a stone whirled with a string.

Step 1: write the given quantities with units and convert them to SI units (km/h to m/s: multiply by 5/18; minutes to seconds: multiply by 60). Step 2: choose the direction of initial velocity as positive. Step 3: pick the equation that contains the three known and one unknown quantity. Step 4: substitute and solve, writing the unit. Remember: for a body starting from rest u = 0; for a body that stops v = 0; for retardation a is negative; for free fall a = g.

Distance, displacement: metre (m). Speed, velocity: m/s. Acceleration: m/s². Time: second (s). 1 km = 1000 m; 1 min = 60 s; 1 h = 3600 s. 1 km/h = 5/18 m/s; 1 m/s = 18/5 km/h = 3.6 km/h. Useful conversions: 18 km/h = 5 m/s, 36 km/h = 10 m/s, 54 km/h = 15 m/s, 72 km/h = 20 m/s, 90 km/h = 25 m/s, 108 km/h = 30 m/s.

Distance (scalar) vs displacement (vector). Speed = distance/time; velocity = displacement/time; acceleration = change in velocity/time. Uniform acceleration equations: v = u + at; s = ut + ½at²; v² = u² + 2as. Distance–time graph slope = speed; velocity–time graph slope = acceleration and area = displacement. Free fall: a = g = 9.8 m/s² downward. Uniform circular motion: constant speed but accelerated because the direction keeps changing. Key terms: rest, motion, reference point, uniform/non-uniform motion, retardation.